Estimate Shipping Country Only in Magento 1.9 – How to Configure

If you want to make it required no matter what, just hard-code it in the template. It should be on the input, not just the label.

<label for="postcode" class="required"><em>*</em><?php echo $this->__('Zip/Postal Code') ?></label>
   <div class="input-box">
      <input class="input-text validate-postcode required-entry" type="text" id="postcode" name="estimate_postcode" value="<?php echo $this->escapeHtml($this->getEstimatePostcode()) ?>" />
   </div>

I think this is a concept thing. Usually a store view only depends on the language and should have basically the same assets. That's why most of the shipping and payment methods can be changed at website level only.
It works without any issues if you change it at store view level.
To allow that you will need an extension that overrides the setting of the active flag for the flat rate. For this create a module. Let's call it Easylife_Flatrate with the following files: app/etc/modules/Easylife_Flatrate.xml - the declaration file

<?xml version="1.0"?>
<config>
    <modules>
        <Easylife_Flatrate>
            <active>true</active>
            <codePool>local</codePool>
            <depends>
                <Mage_Shipping /><!-- should depend on the shipping module -->
            </depends>
        </Easylife_Flatrate>
    </modules>
</config>

app/code/local/Easylife/Flatrate/etc/system.xml - the system->configuration fields

<?xml version="1.0"?>
<config>
    <sections>
        <carriers>
            <groups>
                <flatrate>
                    <fields>
                        <active>
                            <show_in_store>1</show_in_store><!-- override the show_in_store value -->
                        </active>
                    </fields>
                </flatrate>
            </groups>
        </carriers>
    </sections>
</config>

If you need to change other fields as well on the store view level add other tags inside this system.xml files just like <active>.